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 Exploring Pentation - Base e quickfur Junior Fellow Posts: 22 Threads: 1 Joined: Feb 2008 02/22/2008, 12:21 AM Ivars Wrote:[...] I would like to "integrate" (or differentiate) e.g pentation to obtain "slower" operation (or faster). [...]Given any function f(x) that grows asymptotically faster than $e^x$, the derivative f'(x) must necessarily grow faster than f itself. The converse is also true: if f(x) grows asymptotically slower than $e^x$, then f'(x) will grow asymptotically slower than f itself. When this difference is precisely by a factor of x (i.e., $\lim_{x\rightarrow\infty}f(x)/f'(x)=Cx+D$ for constants C≠0, D), then f(x) is a polynomial. There exist functions whose asymptotic ratio with their derivatives lie between 0 and Cx+D... I'll leave this as an exercise for the reader. Since tetration, pentation, and higher grow a lot faster than exponentiation, their derivative must necessarily be faster. Moreover, the difference between the derivative of a pentation and pentation will be much greater than the difference between the derivative of tetration and tetration. « Next Oldest | Next Newest »

 Messages In This Thread Exploring Pentation - Base e - by jaydfox - 12/18/2007, 02:57 PM RE: Exploring Pentation - Base e - by andydude - 12/18/2007, 04:45 PM RE: Exploring Pentation - Base e - by jaydfox - 12/19/2007, 06:01 AM RE: Exploring Pentation - Base e - by Ivars - 01/28/2008, 11:01 AM RE: Exploring Pentation - Base e - by Ivars - 02/02/2008, 05:05 PM RE: Exploring Pentation - Base e - by Ivars - 02/02/2008, 10:50 PM RE: Exploring Pentation - Base e - by GFR - 02/02/2008, 11:01 PM RE: Exploring Pentation - Base e - by Ivars - 02/04/2008, 08:07 PM RE: Exploring Pentation - Base e - by quickfur - 02/22/2008, 12:21 AM RE: Exploring Pentation - Base e - by GFR - 02/04/2008, 09:19 PM RE: Exploring Pentation - Base e - by Ivars - 02/05/2008, 11:25 PM RE: Exploring Pentation - Base e - by GFR - 02/06/2008, 03:01 PM RE: Exploring Pentation - Base e - by Ivars - 02/06/2008, 06:23 PM RE: Exploring Pentation - Base e - by Ivars - 02/07/2008, 09:00 PM RE: Exploring Pentation - Base e - by Ivars - 02/07/2008, 09:30 PM RE: Exploring Pentation - Base e - by Ivars - 02/07/2008, 11:20 PM RE: Exploring Pentation - Base e - by Ivars - 02/08/2008, 10:32 AM RE: Exploring Pentation - Base e - by Ivars - 02/08/2008, 10:51 AM RE: Exploring Pentation - Base e - by Ivars - 02/08/2008, 10:56 AM RE: Exploring Pentation - Base e - by Ivars - 02/09/2008, 11:12 PM RE: Exploring Pentation - Base e - by Ivars - 02/15/2008, 08:24 PM RE: Exploring Pentation - Base e - by Ivars - 03/03/2008, 08:04 PM

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